Line Symmetry vs Rotational Symmetry

Recall that reflections in two intersecting lines give the same result as a rotation about their point of intersection through twice the acute angle between the two lines.  As a consequence, if an object has two lines of symmetry, it must also have rotational symmetry about the point of intersection of those two lines.  However, the converse is not true.  An object can have rotational symmetry, but not line symmetry.  The "windmill" which has 60^o rotational symmetry does not have line symmetry.  A parallelogram has 180^o rotational symmetry (with center the intersection of its diagonals) but no line symmetry:

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